The existence of renormalized solution for quasilinear parabolic problem with variable exponents and measure data

Abstract

In this paper, the study of the existence of a renormalized solution for quasilinear parabolicproblem with variable exponents and measure data. The model is: u_{t}-\text{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)+\lambda\left\vert u\right\vert ^{p(x)-2}u=\mu\text{ } &\text{in}\hspace{0.5cm}Q=\Omega \times ]0,T[,\\u=0 & \text{on}\hspace{0.5cm}\Sigma =\partial \Omega \times ]0,T[, \\u(.,0)=u_{0}(.) & \text{in}\hspace{0.5cm}\Omega, where $ \lambda>0$ and $ T $ is any positive constant, $ \mu\in\mathcal{M}_{0}(Q) $ is any measure with bounded variation over $ Q=\Omega \times ]0,T[ $.